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Abstract
Metasurface based on independent and simultaneous control of near field and far field has significant potential for use in multichannel optics platform devices. However, the previous studies cannot satisfy independent and simultaneous control of near field and far field under a single line source, which made a significant challenge to multichannel optical platforms working in a compact environment. To manipulate effectively and freely the amplitude and phase of transmission under line source, Marius’ law and Propagation phase was introduced on all-dielectric encoding metasurfaces meta-atoms. The Marius’ law and Propagation phase can control the size and rotation angle of meta-atoms to encode grayscale amplitude images and holographic phase images. Finite-difference time-domain simulation results reveal that dual channel metasurface under a single line source achieves the same display effect as the dual channel metasurface under multiple light sources, which proves the feasibility of our studies. Moreover, under different angles of the line source, we encode the near-field binary image by using the degeneracy rotation angle of meta-atoms. Finally, a three-channel metasurface was obtained without affecting the display of the previous two-channel metasurface. As a result, the independent control amplitude, phase, and polarization of the incident light wave were achieved. The proposed metasurface could be applied in creating a multi-channel metasurface optical platform in a compact environment, which has application potential in image displays, optical storage, optical anti-counterfeiting, and information encryption technology.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The optical metasurface based on two-dimensional subwavelength nanostructures can be equivalent to a functional material for point-by-point regulation of the electromagnetic field [1–9]. Traditional holography has many problems, such as double imaging, small field of view, unnecessary high-order diffraction, and noise. The dielectric metasurface can be used as a new coding medium to achieve high resolution, large field of view, and eliminate high-order diffraction [10–12]. The phase adjustment mechanism of dielectric metasurface holography can be roughly divided into the geometric phase adjusted only by the rotation angle of nanostructures [13,14], the Propagation phase adjusted by the size of nanostructures [15–17], and the Huygens phase adjusted by the radius size of cylindrical nanostructures [18]. The hologram reconstructed by metasurface diffraction has a complex relationship with the polarization state, phase, and diffraction distance of incident light. Polarization multiplexing holography [19–21], color holography [22], or three-dimensional holography [23] can be realized by using this property. In addition, metasurface also provides a new way for the storage and display of high-density images, which can be used to produce nano-printed images with high resolution, high fidelity, and high efficiency. Dielectric metasurfaces can realize nano printing by adjusting the influence of the nanostructure's sizes on the incident light response [22], or realize nano printing based on Marius’ law by arranging the rotation angle of nanostructures [24]. In addition, only the light wave with a specific wavelength and specific polarization state incident at a specific angle can display the near field image, which greatly improves the security of image information [25–28]. Metasurface nano printing only acts on the near-field region half a wavelength away from the metasurface and depends on amplitude regulation. By controlling the phase to adjust the diffraction distance, metasurface holography can reproduce the holographic image in a region far greater than half the wavelength. Therefore, only independently adjusting the phase and amplitude of the incident light, nano printing, and holography can be applied to the same metasurface to realize multiple information reuse [29,30]. Information multiplexing based on metasurface has the advantages of small size and high information density, which is one of the most important links in the realization of a new generation of the multi-channel optical platform. For example, when simultaneously and independently controlling the response of a single meta-atom to the amplitude and phase of the incident light, a complex amplitude hologram [23] or a synchronous nano printing hologram [31,32] is realized. The independent amplitude and phase functions are encoded into orthogonal [33] or non-orthogonal [34–36] multi atoms to form a supercell, which is used to realize synchronous polychromatic nano printing hologram or synchronous nano printing hologram. However, there are some problems in the above research, such as multi-light source regulation, ohmic loss, and low utilization efficiency of supercells. Therefore, it is still a challenge to realize high-efficiency and independent information reuse under a single light source. Such metasurfaces can be used to develop complex, multi-channel, and multi-functional optical devices for optical engineering in a compact footprint.
Here, we propose a three-channel dielectric metasurface realized under a single-line polarized light source. The metasurface is composed of a series of high aspect ratio titanium dioxide (TiO2) nanocolumns and TiO2 nanocolumns with different directions and sizes located on the quartz (SiO2) substrate. The Propagation phase and Marius’ law can adjust the size and rotation angle of the metasurface element respectively so that the near-field gray-scale amplitude image and the far-field holographic phase image can be encoded independently at the same time. In addition, the near-field binary anti-counterfeiting image is encoded by using the degeneracy of the rotation angle of the nanostructure. Finite-difference time-domain (FDTD) simulation results reveal that dual channel metasurface under a single line source achieves the same display effect as the dual channel hypersurface under multiple light sources. By adjusting the polarization angle of the line light source, a binary anti-counterfeiting image was observed in the near field. The three-channel metasurface proposed in this paper can simultaneously and independently control the phase, polarization, and amplitude of the incident light source. The three-channel metasurface can be used to create a multi-channel metasurface optical platform in a compact environment, which can be applied to multiple image displays, optical storage, optical anti-counterfeiting, information encryption, and other technologies.
2. Principle analysis
2.1 Principle of the metasurface holographic
We analyzed the optical properties of the metasurface using FDTD method, which is accurate in simulating micron-level metasurfaces. To calculate the metasurface holography, a block iterative holography algorithm was used to calculate the electric field distribution of the ideal metasurface structure. There are several phases of retrieval algorithms, such as Gerchberg-Saxton [37], Fienup Fourier [38], and Yang-Gu algorithm [39]. Because we used FDTD software to simulate the micron-scale metasurface, our optical component size and imaging size were very similar.
where z represents the distance from the metasurface to the diffraction observation screen, (x0, y0) represents a fixed point on the observation screen, and (x, y) represents any point on the metasurface. As shown in Fig. 1, the process is described with the Rayleigh–Sommerfeld diffraction formula.Corresponding to the “RS” in Fig. 1, U(x0, y0) and U(x, y) represent the electric field on the observation screen and metasurface, λ is the wavelength of the diffraction design and $\cos \langle \textrm{n,r}\rangle$ is the tilt factor. If the quality of the reconstructed image meets our requirements, output the target phase distribution map, if not, “RS−1” process carries on.
Fig. 1. Design of metasurface hologram based on GS block iterative algorithm. In reproducing a hologram of a picture, the sampling step is omitted, and the encoding is the stepping process of the phases represented by different metasurface units in the next section. The flowchart in the figure is mainly for the third step optimization process, the final reproduction of this part is what we observed through the electric field monitor in the FDTD software.
Continue to iterate.
2.2 Metasurface control of incident light source phase, amplitude, polarization
The principle of Propagation phase action to line light source can be expressed by the following:
where Δφ represents the generated phase, Δneff represents the equivalent refractive index of the metasurface periodic unit. Δneff is intermediate to the vacuum refractive index and the material refractive index and determined by the length Lx and width Ly of the medium unit. λd represents the designed wavelength, and H represents the height of the nanostructure. As shown in Fig. 2(a), Propagation phase modulation requires appropriate incident wavelength and nanostructure height to meet the phase modulation of 0∼2π.Fig. 2. Metasurface control of incident light source phase, amplitude, polarization. (a) Propagation phase modulation. (b)The rotation angle θ of the nanopillar. (c) Encoding near-field binary anti-counterfeiting image by changing the polarization angle of incident light.
The phase of the metasurface was controlled by adjusting the rotation angle of the anisotropic birefringent nanostructure. This phase mechanism is not sensitive to wavelength and is effective when the incident light is circularly polarized light. As shown in Fig. 2(b), the Geometric phase only depends on geometric angle adjustment, the phase adjustment formula:
where Δφ is the phase generated by the rotation of the anisotropic nanopillar around the horizontal axis by the angle θ.In addition, Marius’ law can regulate the near-field amplitude by adjusting the geometric rotation angle under a line polarized light. After passing through the analyzer, the transmitted light intensity of a beam of linearly polarized light with an intensity of I0 is given.
where I is the intensity of the outgoing light wave and I0 is the intensity of the incident light wave. Tx and Ty are complex amplitude coefficients of the x-axis and y-axis of anisotropic nanostructures.Therefore, the light intensity can be controlled to achieve a continuous grayscale nano printing image in the near-field by changing the rotation angle θ. Because the turning angle is determined by Marius’ law, there are limits to the use of far-field holography if the Geometric phase is to be used to control the metasurface. Therefore, the Propagation phase is used as the main mechanism of phase modulation of incident light. To develop the degree of freedom of polarization of the incident light source, the degeneracy of Marius’ law is used to make each corner have four optional values of the same amplitude. The values are θ, π/2-θ, π/2+θ, and π-θ.
As shown in Fig. 2(c), the red curve is the output light intensity when the incident light source is line-polarized light, and the blue curve is the intensity when the polarization angle of the line-polarized light changes. In the four corners with the same intensity in the red curve, the intensity is high or low in the blue curve. The intensity difference of the corner in the blue curve can be used to encode the binary anti-counterfeiting image. So, the binary anti-counterfeiting image can be obtained in the near field when the polarization angle of the incident light changes.
2.3 Realize the independent regulation of incident light source amplitude, phase, and polarization by the metasurface
The Jones matrix simplifies the study of the impulse response of a single nanostructure [40]. If a nanostructured column with anisotropy is represented by the Jones matrix, it can be expressed by the following:
By rotating the birefringent dielectric cube horizontally at angle θ, the following equation can be obtained.
The input pulse is set to linear light polarized (LP) along the horizontal direction, and the pulse passed through the polarizer along the horizontal direction after passing through the metasurface. Substituting into Eq. (3) and getting:
By calculating Eq. (4), the output pulse can be obtained:
To further simplify the form of the output pulse, we assume that the dielectric nanostructure is a half-wave plate. If the birefringent dielectric cube satisfies the half-wave plate effect φx=φy +π, substitute into Eq. (5) to obtain the final form of the output as follows:
The amplitude of the output pulse can be separately regulated by controlling θ, that is it is not affected by adjusting φx. Finally, we can achieve the near- and far-field functions of the metasurface by independently controlling the amplitude and phase. Amplitude multiplexing can be realized by changing the polarization angle of the incident light source. In the three-channel metasurface design shown in Fig. 3, two-channel amplitude images in the near-field are used to implement the anti-counterfeiting function, continuous amplitude image in the near-field can be achieved nano printing function, and phase diffraction reconstruction image in the far-field is used to implement the holographic function. The three-channel metasurface can be applied to multiple image displays, optical storage, optical anti-counterfeiting, and information encryption technologies.
Fig. 3. Operation schematic diagram of three-channel hypersurface realization. The metasurface is composed of half-wave plate nanostructures of different sizes. It uses linearly polarized light to enter.
3. Finite-difference time-domain simulation
3.1 Simulation of metasurface unit and Preprocessing of metasurface holography
Here, we used the Propagation phase mechanism to cover the φx change between 0 and 2π by selecting eight types of half-wave plates with different phases to achieve metasurface holograms.
Through previous theoretical analysis, one knows that when the φx = φy+π is satisfied between φx and φy, a hologram with φx as a variable at a distance can be observed with a horizontal X-line polarized light incident. TiO2 has good transmission performance in the visible band, therefore we used SiO2 as the base and piled TiO2 nanopillars on SiO2. X-line polarized light with a wavelength of 480 nm was selected as the incident light source, and the refractive index of the SiO2 substrate material and nanomaterial TiO2 at this wavelength are 1.46 and 2.76, respectively. To make φx change from 0 to 2π, a nanopillar with a high aspect ratio is required, therefore the parameters of height H = 600 nm and unit period P = 400 nm are chosen. FDTD software is used to simulate a TiO2 nanopillar in the center of the SiO2 substrate and optimized the structure by adjusting Lx and Ly. As shown in Fig. 4(a), φx can be changed by changing Lx and Ly. The width Ly and length Lx of unit cells from 1 to 4 are depicted as follows: Lx = 210 nm, 208 nm, 230 nm, 278 nm, and Ly = 70 nm, 90 nm, 98 nm, and 110 nm. Units 5-8 can be obtained by swapping the length and width of units 1-4. Define the cross-polarization transmission efficiency as (Tx-Ty)2/4. The cross-polarization transmission efficiency of different units remains similar, so that the amplitude regulated by Marius’ law tends to be homogenized, as shown in Fig. 4(b), Propagation phase can cover 0∼2π. More details of the metasurface unit are shown in Fig. 4(c)
Fig. 4. Metasurface units design and simulation. (a)The simulation of the metasurface unit cell. (b)The cross-polarization transmission efficiency and the Propagation phase of the unit cell. (c) The simulated unit cells detail at the wavelength of 480 nm.
Under the 480 nm X line light source, we used some units of 85×85 form metasurface to calculate the metasurface holographic at the diffraction imaging distance z = 20 µm. As shown in Fig. 5(a), GS iterative algorithm is used to obtain the phase of metasurface holography. First, input the number of metasurface elements, the wavelength of the incident light, and the diffraction distance as light detail into the GS iterative algorithm process. After the calculation of the Rayleigh–Sommerfeld diffraction algorithm, the random phase diagram is superimposed on the target amplitude. If the diffraction reconstruction image meets the required requirements, the target holographic phase is output through the inverse Rayleigh–Sommerfeld diffraction algorithm. If the requirements are not met, the initial value of the next iteration is the holographic phase output by the inverse Rayleigh–Sommerfeld diffraction algorithm.
Fig. 5. Preprocessing of metasurface holography (a) Metasurface phase hologram obtained by GS iterative algorithm. (b) The relationship between the Geometric phase, Propagation phase, and target phase.
However, the influence of the Geometric phase on the metasurface holography cannot be ignored. Considering the phase superposition produced by the Geometric phase, as shown in Fig. 4(b), the phase distribution of the phase hologram is φ, and the Geometric phase difference caused by the rotation angle is 2θ. From the formula in Fig. 5(b), the target phase that needs to be encoded by the size of the metasurface element is Φ.
The target phase hologram is obtained by different transmission phase codes generated by eight kinds of metasurface elements. However, the discrete Propagation phase can’t fully compensate for the holographic phase of the GS algorithm. Therefore, the diffraction reconstruction image obtained by the Propagation phase is 0.92 times of the GS diffraction algorithm
3.2 Simulation of three channel metasurface and numerical analysis
The design steps of three-channel metasurface are shown in Fig. 6. Under line polarized light, the continuous amplitude image and the binary anti-counterfeiting image “华师” are encoded through the corner of each metasurface unit. The continuous amplitude image is the emblem of South China Normal University. The binary anti-counterfeiting image is displayed when the X-line polarized light is deflected by 45 °. Then, the result of the hologram preprocessing is used to encode the size of each metasurface unit through the Propagation phase under a certain rotation angle of metasurface units. After obtaining all the details of metasurface elements, a program was set to code each metasurface element in FDTD software.
Fig. 6. Design description of three-channel metasurface. Input the size and rotation angle of the metasurface element into the FDTD software, and set the light source parameters before simulation
It should be noted that the continuous amplitude image of the near-field is obtained at the plane half a wavelength away from the metasurface, while the amplitude image of the near-field is not observed in the plane much larger than the wavelength. This is the difference between Marius’ law and the Geometric phase. In addition, Marius’ law and the Geometric phase have a difference in acting on different light sources. Marius’ law acts on a line of polarized light sources, while the Geometric phase acts on a circularly polarized light.
In the previous research, through the complete decoupling of near-field and far-field amplitude and phase, a multifunctional hypersurface with continuous near-field amplitude and original focus under dual light sources was realized. [29] Based on the same regulation, near-field continuous amplitude and far-field holography under dual light sources can be realized. Therefore, using FDTD software to simulate the dual channel metasurface based on dual light sources, the dual channel metasurface of a single light source, and the single light source three channel metasurface in this paper. As shown in Fig. 7, comparing the display effect between the dual channel metasurface of a single light source with the dual channel metasurface of dual light sources, it can be proved that the phase deviation caused by the Geometric phase has been eliminated. Therefore, it is not necessary to use circularly polarized light to realize Geometric phase compensation. The dual channel metasurfaces can be realized under a single-line polarized light source.
Fig. 7. Comparison between two channel metasurfaces with the dual light source, two-channel metasurfaces with a single light source, and three channel metasurfaces with a single light source. (a), (b) dual channel metasurface based on a linearly polarized light source and left-handed circularly polarized light. (c), (d) dual channel metasurface based on a single linearly polarized light source. (e)-(g) three-channel metasurface based on a single linearly polarized light source.
As shown in Fig. 7, it can be found that the display effect of the three-channel metasurface proposed in this paper is almost the same as that of the two-channel metasurface of the single light source and the double light source. In addition, the binary anti-counterfeiting image is achieved by deflecting the X-line polarized light source at 45 °, which represents that the three-channel metasurface in this paper realizes the phase, amplitude, and polarization multiplexing based on the independent regulation of near and far fields.
Numerical analysis is used to further study the image display differences between dual channel-single light source, dual channel-dual light source, and three channels. The universality of this difference is proved by changing the number of metasurface element arrays. Based on GS iterative diffraction algorithm, three kinds of metasurface diffraction image recoveries are used to test the excellence of the far-field holographic effect. As shown in Fig. 8(a), the recovery curves of dual channel-single light source and dual channel-dual light source are very consistent. However, when the size of the metasurface is small, the recovery curve of three channel metasurface is slightly inferior to that of the two-channel metasurface. When the size of the metasurface is big, the recovery curve of three channel metasurface is slightly better than that of the two-channel metasurface. In general, there is no big difference between the three kinds of metasurfaces in far-field holographic reconstruction.
Fig. 8. Numerical analysis and three-channel metasurface of optical storage function. (a) GS algorithm recovery curves of three kinds of the metasurface. (b) Continuous amplitude recovery curves of three kinds of the metasurface. (c) three channel metasurface simulation of optical storage function.
However, the three-channel metasurface is significantly better than the two-channel metasurface in the near-field continuous amplitude recovery curve. As shown in Fig. 8(b), the near-field continuous amplitude recovery curve of three channel metasurface is slightly better than that of two-channel metasurface of any size. Therefore, in general, a three-channel hypersurface with higher performance can be achieved based on the two channels. In addition, we can also realize a three-channel metasurface with optical storage function based on the previous three-channel metasurface. As shown in Fig. 8(c), the continuous amplitude is encoded to realize the logo of the “College of information Optoelectronics Technology”. By changing the polarization angle of the incident light wave, the binary image “光电” can be observed. The diffraction reconstruction image of the letter B can be observed in the far field.
4. Conclusion
In summary, we propose a new all-dielectric metasurface three-channel design scheme based on independently regulating the phase, polarization, and amplitude. The metasurface is composed of a series of high aspect ratio titanium dioxide (TiO2) nanocolumns, and the TiO2 nanocolumns with different sizes and rotation angles are located on the quartz (SiO2) substrate. In the design of three channel metasurface element, Marius’ law and Propagation phase are combined, and the Geometric phase deviation is adjusted. By comparing the reconstructed holographic diffraction images of line polarized light source and circularly polarized light, we eliminate the phase deviation caused by the Geometric phase. In addition, polarization multiplexing is realized by changing the polarization angle of linearly polarized light. Finally, we get the third channel that includes a binary anti-counterfeiting images with watermark and binary images without the watermark. The three-channel metasurface proposed in this paper realizes the independent regulation of phase, polarization, and amplitude, and can be used in the fields of optical storage, optical anti-counterfeiting, holographic display, and optical encryption. It is worth mentioning that the three-channel metasurface works under a single linearly polarized light, which provides a new potential choice for multifunctional optical platform equipment working in a compact environment.
Funding
Science and Technology Planning Project of Guangdong Province (2017A020219007); Natural Science Foundation of Guangdong Province (2016A030313443); National Natural Science Foundation of China (1167410961774062).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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